Einstein offered a geometric explanation of gravity. Rather than viewing gravity as an attractive force between massive objects, which is how Newton described it, Einstein’s theory is based on the notion of curvature. The presence of mass, he teaches us, literally causes the fabric of space and time to curve or warp. The warping of space and time affects the motions of objects nearby, and that, in essence, is the phenomenon we call gravity.
To explore this idea, consider two points on the sphere. There is a unique shortest path (or geodesic ) on the sphere between these two points (Fig. 45 ). Although this path is not “straight” in the conventional sense, it is still the shortest path that lies in the sphere. Actually, there are exceptions: there are infinitely many shortest-length paths between antipodal points. Instead we can start from a given point on the sphere and go in a given direction in as straight a manner as we can. This is what is called a geodesic. The path we thus get will be a great circle on the sphere.
Other kinds of situations can arise. On the torus, a donut shaped object as in Fig. 46 , there are topologically inequivalent geodesics paths between points lying on opposite sides of a cross-sectional circle.
